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To determine how many elements in a Postgres array overlap with each other, you can use the OVERLAPS operator along with the array intersection operator &&.

Here's an example query that returns the count of overlapping elements in two arrays:

SELECT COUNT(*) FROM unnest(ARRAY[1, 2, 3, 4]) a
WHERE a = ANY(ARRAY[3, 4, 5, 6]) AND a && ARRAY[3, 4, 5, 6];

This will return a count of 2, indicating that the arrays overlap on elements 3 and 4.

You can adjust the arrays in the query to compare any two arrays and determine the count of overlapping elements.